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3.4. General theory (Davies–Lewis–Ozawa)

Finally, we formulate the general notion of quantum instrument. A superoperator acting in ${\textstyle {\mathcal {L}}({\mathcal {H}})}$ is called positive if it maps the set of positive semi-definite operators into itself. We remark that, for each $x,\Im _{A}(x)$  given by (13) can be considered as linear positive map.

Generally any map$x\rightarrow \Im _{A}(x)$ , where for each $x$, the map $\Im _{A}(x)$ is a positive superoperator is called Davies–Lewis (Davies and Lewis, 1970) quantum instrument.

Here index ${\textstyle A}$  denotes the observable coupled to this instrument. The probabilities of ${\textstyle A}$-outcomes are given by Born’s rule in form (15) and the state-update by transformation (14). However, Yuen (1987) pointed out that the class of Davies–Lewis instruments is too general to exclude physically non-realizable instruments. Ozawa (1984) introduced the important additional condition to ensure that every quantum instrument is physically realizable. This is the condition of complete positivity.

A superoperator is called completely positive if its natural extension ${\textstyle \jmath \otimes I}$ to the tensor product  ${\textstyle {\mathcal {L}}({\mathcal {H}})\otimes {\mathcal {L}}({\mathcal {H}})={\mathcal {L}}({\mathcal {H}}\otimes {\mathcal {H}})}$ is again a positive superoperator on ${\textstyle {\mathcal {L}}({\mathcal {H}})\otimes {\mathcal {L}}({\mathcal {H}})}$. A map $x\rightarrow \Im _{A}(x)$ , where for each ${\textstyle x}$, the map $\Im _{A}(x)$ is a completely positive superoperator is called Davies–Lewis–Ozawa (Davies and Lewis, 1970, Ozawa, 1984) quantum instrument or simply quantum instrument. As we shall see in Section 4, complete positivity is a sufficient condition for an instrument to be physically realizable. On the other hand, necessity is derived as follows (Ozawa, 2004).

Every observable  ${\textstyle A}$ of a system ${\textstyle S}$ is identified with the observable ${\textstyle A\otimes I}$ of a system ${\textstyle S+S'}$ with any system ${\textstyle S'}$ external to${\textstyle S}$ .10

Then, every physically realizable instrument  $\Im _{A}$ measuring ${\textstyle A}$ should be identified with the instrument  ${\textstyle \Im _{A}{_{\otimes }}_{I}}$ measuring ${\textstyle A{\otimes }I}$ such that ${\textstyle \Im _{A}{_{\otimes }}_{I}(x)=\Im _{A}(x)\otimes I}$. This implies that ${\textstyle \Im _{A}(x)\otimes I}$ is agin a positive superoperator, so that $\Im _{A}(x)$ is completely positive.

Similarly, any physically realizable instrument $\Im _{A}(x)$ measuring system ${\textstyle S}$ should have its extended instrument  ${\textstyle \Im _{A}(x)\otimes I}$ measuring system ${\textstyle S+S'}$ for any external system${\textstyle S'}$. This is fulfilled only if  $\Im _{A}(x)$ is completely positive. Thus, complete positivity is a necessary condition for $\Im _{A}$ to describe a physically realizable instrument.